End anchoring in short-term order memory. Simon Farrell and Anna Lelièvre. Department of Psychology. University of Bristol

Transcription

1 End anchoring in STM Running head: END ANCHORING IN STM End anchoring in short-term order memory Simon Farrell and Anna Lelièvre Department of Psychology University of Bristol Draft of September, 008. Dr. Simon Farrell Department of Experimental Psychology University of Bristol a Priory Road Clifton, Bristol BS8 TU UK Simon.Farrell@bristol.ac.uk fax: + (0)

2 End anchoring in STM Abstract Temporally grouping lists has systematic effects on immediate serial recall accuracy, order errors, and recall latencies, and is generally taken to reflect the use of multiple dimensions of ordering in short-term memory. It has been argued that these representations are fully relative, in that all sequence positions are anchored to both the start and end of sequences. A comparison of four computational models of serial recall is presented that shows that the extant empirical evidence does not point towards fully relative positional markers, and is consistent with a simpler scheme in which only terminal items are coded with respect to the end of a sequence or subsequence. Results from the application of the models to data from two new experiments varying the size of groups in serially recalled lists support this conclusion. Keywords: serial recall; short-term memory; temporal grouping; sequence memory; computational models; model selection; anchors.

3 End anchoring in STM End anchoring in short-term order memory Evidence from investigations across psychology, including word recognition (e.g., Davis & Bowers, 00), economic and valuative judgements (e.g., Hsee, Hastie, & Chen, 008; Stewart, Chater, & Brown, 00), spatial representation (Sadalla, Burroughs, & Staplin, 980) and perception and absolute judgement (e.g., Bressan, 00; Gravetter & Lockhead, 9; Stewart, Brown, & Chater, 00), have converged on the conclusion that in many cases the representation or judgement of objects and values is relative in nature. For example, although word recognition models have commonly assumed that letter information is encoded in an absolute, position specific way (Coltheart, Rastle, Perry, Langdon, & Ziegler, 00; McClelland & Rumelhart, 98), recent evidence on letter confusions in briefly presented letter strings shows that transpositions of letters from different positions within words are more frequent than expected from absolute models, implying relativity in the way letters within words are represented (Davis & Bowers, 00). In a similar vein, a striking phenomenon from a number of domains is that of the range effect, whereby participants judgements of the physical properties of objects such as lightness (e.g., Bressan, 00), length (e.g., Lacouture, 99), and sweetness (Lawless, Horne, & Spiers, 000) are shifted according to the range of stimuli to which the observer has been exposed. One common assumption is that participants use some form of anchoring, whereby the representations or judgements of stimuli are anchored to landmarks including the extreme stimuli presented in an experiment (e.g. Braida et al., 98; S. D. Brown, Marley, Donkin, & Heathcote, 008). In this paper, we consider whether a similar principle of anchoring is at play in short-term order memory. Specifically, we ask to what extent the coding of position of items or events is anchored to both the start and end of that sequence (Henson, 998b; Houghton, 990). As we discuss below, this issue is of particular

4 End anchoring in STM significance given one idiosyncratic characteristic of temporal order: the length of a sequence may often be unknown until the entire sequence has been presented, meaning that the end anchor is not usable as a stable referent during encoding. Below, we discuss a number of models of positional representation in short-term order memory, and present a series of simulations and experiments addressing the role of anchoring in short-term order memory. Varieties of positional representations in short-term memory A major focus of models of short-term memory has been the representations or mechanisms by which the order of sequences is remembered. One class of models, primacy models, assumes that items are represented by a gradient in the strength of activations or associations across a sequence, such that earlier items are more accessible (Farrell & Lewandowsky, 00; Page & Norris, 998). This simple ordinal scheme has been demonstrated to be sufficient to account for a number of benchmark data from the serial recall task, including serial position effects in accuracy and latency, the locality in positional confusions (nearby items tend to be confused), list length effects, and other phenomenon such as modality and word frequency effects (Farrell & Lewandowsky, 00, 00; Page & Norris, 998). However, there exist data which are incompatible with the basic gradient-based representation assumed in these models. One major objection to primacy models comes from the multifarious effects of grouping on serial recall performance. Grouping a list into subsequences by inserting pauses between groups (e.g., Maybery, Parmentier, & Jones, 00; Ryan, 99a, 99b), intonation (Frankish, 99; Reeves, Schmauder, & Morris, 000), or simply suggesting a grouping structure through verbal instructions (Farrell, 008; Wickelgren, 9) lead to a number of well-replicated effects on recall. When examining accuracy, grouping leads to the appearance of mini serial position curves for groups, each group with its own

5 End anchoring in STM primacy and recency. Grouping also has systematic effects on recall latencies: participants leave longer pauses in their output at group boundaries (Farrell, 008; Maybery et al., 00). Most problematic for primacy models is the effects of grouping on recall errors, particularly those involving confusions of items between groups (Henson, 99). When adjacent positional confusions are examined, confusions between groups tend to dominate for ungrouped lists, while confusions within groups dominate for grouped lists (e.g., Maybery et al., 00). More tellingly, grouping lists increases the tendency of participants to produce interpositions: if an item is recalled in the incorrect group, it nevertheless tends to be recalled at the correct-within group position. For example, in a -item list grouped into two -item groups, the th item (i.e., the second item in the second group) will, if recalled anywhere in the first three positions, tend to be recalled at the second position (that is, the second position in the first group; e.g., Henson, 99; Lee & Estes, 98). The pattern of data arising from grouping is inconsistent with primacy models because those models predict that primacy will dominate in any confusions between groups (e.g., an anticipation of an item from a later group will always tend to involve the first item from that group). Although it might be argued that positional confusions occur in some other optional mechanism separate from that accounted for with a primacy gradient (Page & Henson, 00; Page & Norris, 998), this weakens these models as universal models of short-term order memory. Alternatively, these data have been taken to indicate a role for positional representations in short-term memory. A number of models assume that the order of items is stored by associating each item with a positional marker specifying the position of the item. By assuming that positional markers for nearby positions overlap, models incorporating positional representations are able to account for a large range of serial recall data (J. R. Anderson & Matessa, 99; G. D. A. Brown, Neath, & Chater, 00; G. D. A. Brown, Preece, & Hulme, 000; Burgess & Hitch, 999; Henson, 998b; Lewandowsky & Farrell, 008b). In particular, these

6 End anchoring in STM models can account for the interpositions seen in grouped lists by assuming representations for both the position of items in a group, and a coarser representation of the items position in the list as a whole (item-in-list: G. D. A. Brown et al., 000; Burgess & Hitch, 999) or the position of the group in the list (group-in-list: J. R. Anderson & Matessa, 99; Henson, 998b; Lewandowsky & Farrell, 008b). These hierarchical representations are of apparent generality, as participants may spontaneously group lists even when those lists are presented as homogeneous structures (e.g., Madigan, 980). Although phenomena such as grouping effects do seem to mandate some positional representations, it could be argued that incorporating such representations shifts the burden of explanation for ordering in the first place. In other words, if ordering is not a property of the items but of some external mechanism, how is that mechanism itself able to correctly order and retrieve its positional representations? The most basic answer to this question is found in models such as that proposed by Conrad (9), who suggested that items in a sequence are stored in ordered bins in memory. Contemporary models have replaced this scheme with more detailed specifications of the generation of positional representations and their functional relationship across positions (G. D. A. Brown et al., 000; Burgess & Hitch, 999; Henson, 998b). For example, an appealing mechanism for the generation of positional representations is found in the oscillator-based model of G. D. A. Brown et al. (000), which assumes that a bank of time-varying sinusoidal oscillators generates a temporal context signal that is used to distinguish elements of a sequence and support ordered recall. In the case of grouped lists, an additional bank of oscillators is assumed to be used to code within-group position (see also Burgess & Hitch, 999; Hitch, Burgess, Towse, & Culpin, 99). These models, whether they specify the positional representations as varying over time (G. D. A. Brown et al., 000) or position (Burgess & Hitch, 00; Lewandowsky & Farrell, 008b) can generally be classified as absolute models (Henson, 999a) or start-anchor models: items are represented by their

7 End anchoring in STM position (or time) since the beginning of a group or list. This representational scheme can be contrasted with a relative coding scheme, in which items are anchored to both the start and the end of a sequence or subsequence (cf. Braida et al., 98). The clearest demonstration of this scheme has been in the Start-End Model (SEM) of Henson (998b), based on the work of Houghton (990). In SEM it is assumed that encoding the order of items consists of the storage of episodic tokens. These tokens contain information about the elements of a sequence, and also incorporate positional representations. Positional information is jointly represented by the values of a start marker and an end marker. The value of the start marker starts at some maximal value and declines across positions, thus representing proximity to the beginning of the (sub)sequence. In a complementary fashion, the end marker increases in value across positions, reaching some maximal value at the end of the sequence. When both these markers are used to represent position (see, e.g., bottom left panel of Figure ), sequence elements are represented with respect to both the start and the end of a list. In the case of grouped lists, this relative marking scheme also applies to representation of position within groups. Henson (999b) presented empirical evidence for the relative marking scheme in short-term order memory. Henson ran an experiment in which participants studied -item lists for serial recall. The lists were presented ungrouped; with a pause between the third and fourth items ( grouping); or with a pause between the fourth and fifth item ( grouping). Of crucial interest was the frequency of interposition errors between positions sharing absolute within-group position, and those sharing relative within-group position. For example, in the case of the grouping, a confusion involving the third and sixth list positions would be considered an absolute positional error, since both list positions share a within-group position of (from the beginning of the group). On the other hand, a confusion of the third and seventh list items would be considered a relative error, since

8 End anchoring in STM 8 both positions correspond to the last item in their respective group. Henson (999b) showed that relative errors dominated in such comparisons, even when the possibility of guessing strategies was addressed by removing reported guesses. Along with evidence that erroneous recalls of items from preceding lists (called protrusions) also tend to follow relative position, these data are consistent with the relative marking scheme assumed in models such as those of Henson (998b) and Houghton (990). Although this explanation and the data offered for it are appealing, the start-end scheme comes with some theoretical baggage. As noted earlier, anchoring items to the end of a sequence presents a substantial predicament: in contrast to other domains in which such anchoring has been proposed (e.g. Braida et al., 98; S. D. Brown et al., 008), end-anchoring in along the temporal dimensions requires anchoring with respect to an unexperienced and unknown referent. One suggestion is that participants form some expectation about the end of the list, and use this expectation to inform the placement of the end anchor (Henson, 999a). Although this sounds reasonable, this explanation is problematic in the case where the end of a sequence cannot be anticipated, such as when a variable list length is used in experiments (Henson, 999a). Although the accuracy of serial recall is reduced under such conditions (e.g., Crowder, 99; Henson, 99, 999b), participants are nonetheless able to recall such lists and display a normal recency effect. It is not clear in such a case how an end anchor could continue to be used without having severely detrimental effects on performance. In the specific case of Henson s (998b) model, in which the end marker increases exponentially, under-estimating the length by only one or two positions may wreak havoc, as the end marker will continue to increase exponentially to capture the unexpected items at the end of the sequence. One possibility, explored in simulations by (Henson, 99), is to have the end marker grow in a fixed fashion and then level off at a fixed upper limit to account for under-estimation of sequence length. Although this limits pathological behavior in the model, in fits presented

9 End anchoring in STM 9 by (Henson, 99, p. 8) the model predicts that the effects of variability in list length are restricted to the final list items, whereas the data show that list length variability affects accuracy at all internal positions (Henson, 99, p. ). Other mechanisms have been presented for circumventing the problem of the unknown end marker, particularly in cases where the end of a sequence or subsequence cannot be anticipated (Henson, 99; Henson & Burgess, 99; Houghton, 990; see Henson, 999a, for a discussion). For example, Henson and Burgess (99) suggested that a number of oscillators of different frequency are initiated at the beginning of a (sub)sequence, and that the oscillator whose period gives the best match to the time scale of the sequence is then used to store the positions in that sequence. Effectively, his mechanism circumvents the problem of incorrectly estimating the length of an upcoming sequence by assuming that a large number of possible sequence lengths are considered in parallel. This mechanism is unappealing in introducing some redundancy into positional models, as each list item must be associated to all oscillators at the time of its presentation. In addition, since the oscillators are time-based, this mechanism cannot account for the insensitivity of serial recall accuracy to timing variation within lists (e.g. Lewandowsky & Brown, 00; Lewandowsky, Brown, Wright, & Nimmo, 00); although one explanation might be that the temporal gaps are taken to indicate the ends of groups and affect parsing of the list, the evidence suggests that such a grouping strategy cannot account for this lack of effect (Lewandowsky et al., 00). The serial recall of grouped lists with irregular timing of items within the groups, but where overall group duration is held constant, is similarly insensitive to within-sequence timing (Ng & Maybery, 00). Although the results from the experiments of Henson (999b) constitute evidence against absolute positional models, they do not necessitate the assumption of fully relative positional markers in short-term memory; not all items need be anchored to the end of the sequence. As noted by Henson (999b), these results are equally compatible with a model

10 End anchoring in STM 0 in which only the last item in a group is represented with respect to the end of the group, as the critical comparisons in Henson s design all involve terminal items. A similar point was made by Page and Norris (998) in their consideration of how grouping effects might be explained in a primacy model. Page and Norris argued that a primacy gradient was sufficient to account for the majority of phenomena in serial recall (see also Farrell & Lewandowsky, 00), but that this gradient might be supplemented by markers representing the start, middle, and end of groups for supra-span lists (Page & Henson, 00). In the case of groups larger than three items, all internal items might be associated with middle markers, with the first and last items being respectively associated with start and end markers. In summary, the existing data suggest some form of end anchoring occurs, but does not specify how extensive this end anchoring is. Is only the last item on the list anchored to the end, or do we need to consider more complicated and theoretically burdensome explanations associated with assuming that all items are end-anchored? In the following, we use model selection to determine whether the extra complexity introduced by assuming that all items are anchored to the end of a sequence is warranted by the existing data (Henson, 999b). Modeling of evidence for relative positional representations As a first step, we revisited the data of Henson (999b) to determine whether the existing grouping data are sufficient to discriminate between end anchoring restricted to terminal items, and extensive end anchoring applying to all items. A set of simulations was conducted in which four models were fit to the data from Experiment of Henson (999b), which manipulated the grouping pattern of grouped lists (ungrouped, or grouped in a or fashion). The models, which incorporate different assumptions about the nature of positional representations in grouped lists in line with the preceding discussion,

11 End anchoring in STM are illustrated in Figure. Start-only model The base model, termed the start-only model, assumed that all items in a group were anchored to the start of the group only. As illustrated in the top-left panel of Figure, this was accomplished by adopting the start marker from the model of Henson (998b), with the strength of the start marker dropping across positions within the group. Although this appears similar to a primacy gradient model, items were still associated with positional representations (the single value of the start marker), such that items were cued by presenting a start marker value and matching this to the start markers of stored items; the effect of the decelerating decrease was to render items at the end of a list less distinct than those at the start. This model represents an absolute encoding scheme equivalent to the positional scheme assumed in a number of models of serial recall (Burgess & Hitch, 999; G. D. A. Brown et al., 000; Lewandowsky & Farrell, 008b); the decelerating function captures the primacy within groups. Following Henson (998b), it was assumed that a start marker was additionally used to represent the position of groups in the list as a whole. Preliminary simulations suggested that inclusion of an end marker for representing group-in-list position did not make any substantial contribution to the performance of the model; a similar observation was made by Henson (998b), on which basis a group-in-list end marker was not included. Restricted end model The restricted end model builds on the start-only model by introducing an additional end marker that is associated specifically with the last item in a group. This is illustrated in the top right panel of Figure. All group items are associated with the continuously decreasing start marker; the end marker is only turned on for the last item in the group, such that only that item is anchored to the end of the group. The difference

12 End anchoring in STM between the restricted-end model and the start-only model tells us about the additional contribution of any end anchoring. This model is in principle sufficient to handle the data of Henson (999b), which deals only with relativity in interpositions involving terminal items. Extensive end model The extensive end model generalizes the restricted end model by allowing the end marker to continuously increase across the group to some maximal value in the same manner as the start marker (see bottom left of Figure ). This scheme, which directly instantiates the positional markers assumed in the start-end model of Henson (998b), means that all items are anchored to both the start and the end of the group. The difference between this representation scheme and the restricted end scheme addresses the central question posed here: How continuous are relative representations in short-term memory? Primacy + start-middle-end The final model considered here is an implementation of a variant of the restricted end marking scheme that is intended to be complementary to primacy gradient models. As suggested by Page and Norris, it is assumed that the position of items within groups is coded by generic labels for the start, middle, and end of the list. The operation of this scheme is depicted in the bottom-right panel of Figure. For the first position in the group, the start marker is activated, with the middle and end markers off. For medial positions, only the middle marker is activated, and for the terminal group item the end marker is solely activated. Additionally, the order of the items within the list as a whole is represented by a unidimensional primacy gradient of activation across all list items (not pictured in Figure ), as assumed in primacy gradient models (Farrell & Lewandowsky, 00; Page & Norris, 998).

13 End anchoring in STM Model implementation The four models were fit in a common framework, in a similar spirit to the lateral inhibition framework for modeling serial recall latencies employed by Farrell and Lewandowsky (00). The purpose of this framework was to allow the comparison of the representational assumptions under controlled conditions. By keeping everything constant except for the changes of interest in the representations (with one exception for the primacy gradient model; see below), any changes in the behavior of the model can be uniquely attributed to the changes in representations. It might be objected that implementing these principles in such a framework may not reflect their true behavior; however, it is shown below that the behavior of the models is exactly as is would be predicted from their representational mechanisms. The simplicity of this framework necessitates a tight linkage between the representational assumptions and the predictions of the resulting model, meaning that we can with confidence reason from the model predictions back to the underlying representations. For the start only model, restricted end, and extensive end models, items were paired with a vector of markers p = {S, E} representing the position of the item in the group, and an additional marker G representing the position of the group in the list. Following Henson (998b), S and E were assumed to vary exponentially across the list, with the value of the markers for the jth item in a group given by S(j) = α j () and E(j) = ω 0 ω N i j, () where N i is the length of the group i containing the item, and α was a free parameter. An

14 End anchoring in STM exponential relationship between distance and contextual overlap was also assumed by Farrell and Lewandowsky (00) in their implementation of a number of a number of serial recall models in a common modelling framework, and is consistent with the fall-off in similarity with increasing distance in the models of Lewandowsky and Farrell (008b) and G. D. A. Brown et al. (000). The parameters ω 0 and ω were used to build up the start only, restricted end and extensive end models. For the start only model, ω 0 and ω were set to 0, meaning E was equal to 0 for all group members. For the restricted end model, ω was fixed at 0 and ω 0 was allowed to freely vary; this had the effect of setting all group items to 0, except for the last group member E(N i ) which had a value of ω 0. Finally, for the extensive end model, both ω 0 and ω were left as free parameters, giving a continuous exponential across all group members. The position of the group i in the list was given by a single group marker G, also varying exponentially: G(i) = ρ i () where ρ was a free parameter. Following Henson (998b) and Burgess and Hitch (999), ungrouped lists were treated as lists containing a single group: group-in-list markers G were set to be identical (equal to ), and thus did not distinguish between list items. Recall in these three models was enacted as in positional models of serial recall (G. D. A. Brown et al., 000; Burgess & Hitch, 999; Henson, 998b; Lewandowsky & Farrell, 008b), by stepping through the positional representations in the same order as at presentation, and using the positional markers corresponding to each position to cue for an output. At each position, all list items were activated to an extent based on the overlap between their associated positional markers, and that corresponding to the position currently being cued. The overlap o I (c, l) between the positional marker of the cued position c, p(c), and the list item l, p(l), for the item-in-group markers was given by

15 End anchoring in STM o I (c, l) = p(c) p(l) exp (p k (c) p k (l)), () where indicates the inner product between vectors (see J. A. Anderson, 99) and k indexes the start and end components within the vector p. Similarly, an overlap in the markers denoting group-in-list position was given by k o G (c, l) = G(c)G(l) exp ( G (c) G (l) ), () which is a simplified version of Equation, given G is a scalar. Following Henson (998b), these overlaps were used to calculate an activation value for each item l in response to cue c as a(l) = o I (c, l)o G (c, l)( r(l)), () An item was selected for recall using a probabilistic selection procedure. For the first set of simulations where group data were fit, zero-mean Gaussian noise with standard deviation σ c (a free parameter) was added to the activations, and the item with the highest noisy activation was selected for response (see, e.g., Henson, 998b; Lewandowsky & Farrell, 008b; Page & Norris, 998). This item was then suppressed by limiting its activation at following output positions (see, e.g., Farrell & Lewandowsky, 00; Henson, 998a; Lewandowsky, 999; Vousden & Brown, 998): formally, r(l) in Equation was set to.9 (Henson, 998b). Response suppression is assumed in most models of serial recall, and is critical for preventing excessive perseverations in the output of the model (Henson, 998a; Vousden & Brown, 998). The primacy gradient model with start-middle-end marking (primacy+s-m-e) was implemented in a similar fashion to the other models. The primacy gradient across list items (for both ungrouped and grouped lists) followed an exponential function

16 End anchoring in STM S(l) = α l () where l indexes position of the item in the list. Additionally, a vector of markers p = {S, M, E} was assumed to code for position-in-group. As per Figure, for each position all elements were set to 0 except for that element coding the position. For the first item in a group, p = {τ, 0, 0} (i.e., only the start marker was turned on); for medial items, p = {0, τ, 0} (i.e., only the middle marker was turned on); and for the final item in a group, p = {0, 0, τ}. The free parameter τ (0 τ ) was used to effectively weight the contribution of this within-group positional representation. Recall approximately followed the procedure in the other three models. The overlap in the within-group positional markers between the cued position c and list item l was given by o I (c, l) = exp (p k (c) p k (l)), (8) k indexing the start, middle and end elements of vector p. To implement a primacy gradient, an overlap o G (, l) was calculated from the positional marker for item l and the positional marker for the first item: k o G (c, l) = exp ( S () S (l) ), (9) regardless of the actual serial position being cued. This is consistent with the formulation of a primacy gradient as a gradient in the weights connecting list items to a chunk node, which has been implemented in a number of suggested primacy gradient models (e.g., Grossberg, 98; Page, Cumming, Norris, Hitch, & McNeil, 00). The activation was then calculated as per Equation, and the item suppressed by setting r(l) to R init ; R init was left as a free parameter given the critical role of response suppression in

17 End anchoring in STM primacy models, leaving the primacy+s-m-e model with as many free parameters as the restricted end model. Relation between models The models all rely on the same fundamental equations, with the primacy+s-m-e model using a slightly different matching equation given it is driven by a primacy gradient rather than by positional representations. The start-only, restricted end and extensive end models form a series of nested models that directly address the role of various types of anchoring. The start-only model is representative of a number of models of serial recall that implement an absolute representational scheme (e.g. G. D. A. Brown et al., 000; Burgess & Hitch, 999; Lewandowsky & Farrell, 008b). Although these models are already effectively rejected by the data of Henson (999b), no fits have been reported in the literature showing that these models cannot account for those data. If these models, as represented by the start-only model, cannot account for the data (as we would expect), implementing the start-only scheme here allows us to quantify the deviation of the theory from the data; this is essential given the effects in Henson s data are numerically small. The restricted end model generalizes the start-only model by introducing an end marker restricted to the last position in a group. In turn, the extensive end model generalizes the restricted end model by allowing the end marker to extend into the entire group, and play a role in representing the position of all list markers; this model effectively implements Henson s (998b) SEM model. The comparison between these models directly addresses the nature of anchoring in positional representations in short-term memory. Since these are nested models, the contribution of the restricted and extensive end markers is seen by comparing them with the start-only model and the restricted end marker models, respectively, and looking at the differences between the models. If, for example, the extensive end marker introduces

18 End anchoring in STM 8 no benefit in accounting for the data, the extensive end marker model will behave in an identical fashion to the restricted end marker model. If the extensive end marker does introduce some important features into the model, the behavior of the model should change in a systematic fashion that better matches the patterns of positional confusions in the data. Finally, the primacy+s-m-e model addresses the plausibility of combining a primacy gradient with discrete within-group positional markers. Modeling The target data from the experiment of Henson (999b) were contained in the transposition matrix for each grouping condition. For each matrix, the rows correspond to output positions, and the columns correspond to the possible input positions (i.e., position on the input sequence) that may have been recalled. The numbers in each cell of the matrix give the total number of times (aggregated across participants) that the item from each input position was recalled at each output position; the diagonal of this matrix corresponds to correct recalls. The models were fit to the data of Henson (999b) using maximum likelihood (ML) estimation. For a particular set of parameter values (and thus the expected probabilities produced by the model), the likelihood of the observed frequencies for a particular row (output position) is given by the multinomial likelihood function. The log-likelihood was summed across output positions and grouping conditions, and converted to a negative value to be used by the function minimization algorithm of Nelder and Mead (9). For this first simulation, model predictions were obtained from 0,000 trial simulations. To maximize the chance of finding the global minimum for the negative log likelihood for each model, a number of starting points were used for the minimization routine. For all simulations, these points were constructed by selecting three values covering a reasonable range along each parameter, and factorially crossing the values from different parameters.

19 End anchoring in STM 9 The minimized log-likelihoods were converted to Bayesian Information Criterita (BICs; Schwarz, 98), which adjust the fit of the model to account for the bias introduced by using the same data to find parameter estimates and obtain a goodness-of-fit value. Formally, the BIC for a model is given by BIC = ln l + ln N, (0) where ln l is the log of the maximum likelihood, and N is the number of data points (i.e., a measure of degrees of freedom) used to fit the data. Before looking at the fits in detail, the predictions of the models are shown in Figures and under the maximum likelihood (ML) parameter estimates. Figure shows the serial position functions (SPCs) predicted by the models, with the SPCs for the data shown in the first (top left) panel. Figure breaks the predictions down in finer detail by showing the fits of the models to the transposition gradients. Each panel in Figure shows the difference between the transposition matrices predicted by the models, and those calculated from the data of Henson (999b). The columns of matrices correspond to conditions, and the rows correspond to the four different models. In each panel, the difference between the observed and predicted frequency is shown for each combination of input position (i.e., position on list) and output position (where the item was actually recalled). For each combination of output position and input position, a circle indicates the size (diameter of the circle) and direction (cross: positive difference; circle: negative difference) of the difference between the model predictions and the data. For the grouped conditions, dashed lines demark the group boundaries suggested by the temporal gap. The BIC for the start-only model was 9., which is much larger (therefore indicating a worse fit) than the restricted end marker model, with a BIC of.99; this BIC difference equates to extremely strong evidence in favor of the restricted end model. This shows that the addition of an end marker, albeit restricted to the final item,

20 End anchoring in STM 0 significantly improves the fit of the model. The second and third panels of Figure show that both models do an adequate job of accounting for the data, but that the start only model tends to produce insufficient recency within groups, and in the list as a whole. Inspection of Figure shows that the systematic difference between the restricted end and start only models is the underprediction by the start only model of the frequency of correct responses (on the diagonal) at the end of groups, and the overprediction of confusions between the last two items in groups (e.g., excessive confusions of items and in lists grouped in a fashion). Critically, the extensive end model, with a BIC value of 8.0, was not superior to the restricted end model. The difference in BICs between the restricted end and extensive end models equates to a likelihood ratio of. in favor of the restricted end model. That is, when the fit of the models and their number of parameters is taken into account, the restricted end model is. times more likely to have generated the data. The evidence in favor of the restricted end model arises from a lack of difference in the models in terms of log-likelihood, and a heavier penalty applied to the extensive end model due to its extra parameter. The effective lack of difference between the models is demonstrated by the visually identical appearance of their predictions in Figures and. Finally, the primacy+s-m-e model had a BIC value of This indicates a superior in fit to the start-only model, but is far removed from the restricted and extensive end models in terms of goodness of fit. The SPCs predicted by the model in Figure suggest that the primacy gradient in the model produces an excessively monotonic decrease in accuracy across serial positions. The transposition matrices in Figure suggest that the difference between the primacy+s-m-e and the other models comes from its overprediction of confusions between items internal to groups of items (i.e., confusions between items and in the grouping condition, and between and in the grouping condition).

21 End anchoring in STM Together, these results show that, for the data of Henson (999b), anchoring all items to the end of a sequence (i.e., assuming a fully continuous end marker) is not necessary to enjoy the benefits of the end anchor. From the existing evidence, there is no evidence for an extensive end marker over and above a discrete end marker applying only to the terminal item. In the following, two serial recall experiments are presented that provide further evidence on the nature and role of the end marker in serial recall. Experiment examines the confusion of items internal to groups in more detail using a grouping condition. Experiment presents a within-participants replication of previous work by combining a grouping condition (Experiment ) with and grouping conditions (Henson, 999b). The experiments are complemented by simulations quantifying the contribution of an extensive end marker in serial recall. Experiment The aim of Experiment was to provide conditions favorable to the detection of an extensive end marker in serial recall of grouped lists. The major distinction between the restricted and extensive end marking schemes is that in the latter, the end marker covers all within-group positions, including positions internal to groups. Accordingly, an extensive end marker should assist in distinguishing between these internal positions. On this basis, Experiment examined the positional errors arising from grouping 8-item lists in a fashion, which allows for the examination of confusions internal to both groups, and confusions between groups involving internal items. Method Participants. Thirty undergraduate students from the University of Bristol participated in exchange for course credits. All participants were native or fluent English speakers. Each participant provided data for both conditions in the experiment.

22 End anchoring in STM Materials and apparatus. Lists were random permutations of the set of consonants H, K, L, M, P, Q, R, and S. Eighty lists were constructed for each condition (ungrouped and grouped) subject to two constraints: consecutive letters from the English alphabet could not appear in successive positions on a list (e.g., K and L could not appear in successive positions); and an item could not appear in the same serial position on consecutive lists. The experiment was controlled by a PC that presented all stimuli (on a monitor) and collected and scored all responses using the Psychophysics Toolbox for MATLAB (Brainard, 99; Pelli, 99). Procedure. Participants were tested individually in a laboratory. Each trial in the experiment began with a fixation point (a cross) being presented in the centre of the screen for 000ms. This was followed by a blank screen with duration 00ms, which was then followed by presentation of the memory list. Letters were presented one by one on the screen, each letter after the first immediately replacing the preceding item. Each letter was presented for 00ms, with a blank screen lasting for approximately 00ms being inserted between each item. Following presentation of the list there was another blank screen of 00ms, followed by presentation of the cue RECALL on the screen. On presentation of the recall cue, participants were to recall the letters from the list in the order they were presented by typing them on the keyboard. In the first session of the experiment, eighty lists were presented under the conditions described so far (ungrouped lists). In the second session of the experiment, which followed the first session after a break of at least 0 minutes, participants were presented with temporally grouped lists. At the beginning of the second session, participants were informed that the eight letters would now be presented in two groups of four, and were instructed to think of the letters in two distinct groups rather than a single list. An additional pause of 00ms was inserted between the fourth and fifth list items.

23 End anchoring in STM The grouping instruction and additional temporal pause were introduced in the second half of the experiment for all participants. Although this leads to an order confound, this constant ordering of conditions was chosen because the grouping strategy was otherwise expected to continue once people had been presented with grouped lists, and thus contaminate the ungrouped condition (Farrell & Lewandowsky, 00; Henson, 999b). Results Although the main purpose of the data collection was to provide data for modeling, standard analyses are presented to show that the expected effects of grouping hold for these data. Figure shows serial position curves (SPCs) for accuracy (left panel) and latency (right panel). For the accuracy SPC, a (grouping: ungrouped vs grouped) x 8 (serial position) repeated measures analysis of variance (ANOVA) revealed a main effect of serial position [F (, 0) = 0., p <.00] and a main effect of grouping [F (, 9) =., p <.00], with grouped lists being more accurately recalled than ungrouped lists. The interaction between grouping and serial position was also significant [F (, 0) = 8., p <.00]: this is consistent with the scalloped appearance of the SPC in Figure for grouped lists, suggesting primacy and recency within groups (Hitch et al., 99). There does appear to be some non-monotonicity in the ungrouped condition suggesting some spontaneous grouping; this possibility is addressed further in the modelling. Standard effects of grouping were also observed for the latencies (right panel of Figure ), as measured by the time between successive keypresses (or between the recall cue and the first keypress in the case of the first item). A (grouping: ungrouped vs grouped) x 8 (serial position) repeated measures ANOVA revealed a main effect of grouping [F (, 9) =., p =.00], with grouped lists giving shorter recall times on average (mean of 0. ms for ungrouped lists vs mean of 88.0 ms for grouped lists). A

24 End anchoring in STM significant effect of serial position [F (, 0) = 9.0, p <.00] was also observed. The interaction between grouping and serial position was also significant [F (, 0) =.8, p <.00]; consistent with previous observations (e.g., Farrell, 008; Maybery et al., 00) a discrete peak in the latency SPC was observed for grouped lists at the group boundary (between list items and ), again indicative that standard grouping effects were obtained. Modeling The four models were fit to the data from Experiment following the procedure used above to fit the data of Henson (999b), with two exceptions. The first exception was a change in the modeling methodology to take advantage of the fact that individual responses were collected and retained in Experiment. The data were fit to individuals, rather than groups, given the possibility that a summed transposition matrix may not be representative of individuals contributing to that matrix (see, e.g., Hintzman, 980); this was not possible for the data of Henson (999b) as they were provided as an aggregate across participants. In obtaining fits for Henson s data reported above, it was observed that a large number of model runs were required to obtain reasonably stable probability estimates, which would preclude fitting the data of individual participants using the same procedure. To address this issue, an analytic version of the models was used, in which the noisy selection process was replaced by a generalized version of the Luce-Shepard choice function (Ashby & Maddox, 99; Luce, 9; Shepard, 9): p(l) = a(l)γ a(i) γ, () with a free parameter γ replacing the noise parameter from the earlier model. For a given set of activation values, Equation gives the probability of recalling each item in that set. Henson (998b, Footnote ) and Page (000) have suggested that the behavior of this choice function closely captures that of the noisy selection procedure with Gaussian noise. i

25 End anchoring in STM A likelihood was calculated for a participant by obtaining a likelihood for each response based on the events preceding that recall on that trial. For the first output position on a trial, no items have been suppressed and the activation values are directly given by Equation with all r(l) = 0. Following the first response, the item l that was actually recalled by the participant was then suppressed in the model by adjusting r(l). The activations were then calculated for output position to give a likelihood, and so on across all output positions. In the rare case of an omission or intrusion error, the response was ignored (no response suppression was applied, and no likelihood value was calculated). This procedure allows for fast and accurate maximum likelihood estimation for individual participants. The second exception was to address the apparent spontaneous use of grouping in the ungrouped condition evident in the SPCs for Experiment (Figure ). To address this possibility, it was assumed that each response in the ungrouped condition followed from the use of the positional markers for a grouped list with probability mix, and that the positional codes assumed for ungrouped lists were otherwise used with probability mix. For all models the introduction of this parameter was found to significantly improve fit, and so it is presented here. The conclusions are unchanged if participants are assumed to always treat ungrouped lists as ungrouped (i.e., mix is set to 0). Given the examination of fits for individuals in these simulations, the fitting results are presented in a slightly different format. Table gives the results of the data fitting based on model weights. These weights are obtained by converting the corrected log-likelihoods (BICs) into likelihoods, and then scaling these so they sum to ; formally: w i = e.bic i j e.bic j, () where w i is the weight for model i. When applied to the BIC values, the BIC weights can be treated as the probability that the model i is the best model for the data, and thus give

26 End anchoring in STM a measure of the strength in favor of one model in the context of a set of models (Burnham & Anderson, 00; Wagenmakers & Farrell, 00). Table shows that the restricted end model (mean ML estimates: ˆα =.; ˆρ =.; ˆω 0 =.9; ˆγ =.9; mix =.) has a distinctly greater mean weight than the other three models. Compared to the start-only model (mean ML estimates: ˆα =.9; ˆρ =.; ˆγ =.9; mix =.), the restricted end model predicts more recency within groups (Figure ). Figure shows that, as for the data of Henson (999b), the major difference between the restricted end and start-only models in their transposition matrices is in the confusions involving the final and penultimate items in lists: the restricted end model predicts fewer confusions between list positions and, and more correct responses for the fourth item. Figures and show that the restricted end and extensive end models (extensive end mean ML estimates: ˆα =.; ˆρ =.; ˆω 0 =.9; ˆω =.; ˆγ =.; mix =.) make almost identical predictions: the extensive end anchoring in the extensive end model appears to add little explanatory power to the model, which agrees with the much smaller number of wins and lower average BIC weight for that model in Table. Although the extensive end model does provide the best corrected fit in four cases, there are nearly as many wins for the start only model which assumes no end anchoring at all. Nonetheless, the average weight for the extensive end model is superior to both of the other two models. Perhaps some participants are relying substantially on extensive end anchoring? An indication that this isn t the case is seen in Figure, which plots the anchoring strength of internal items to the sequence ends obtained from the extensive end model. That is, by calculating the anchoring strength of the start marker to its adjacent internal item and comparing this to the anchoring strength of the end marker to its adjacent internal item, we can estimate the relative extent to which the end marker protrudes into the list, and thus determine the overall contribution of the end marker. The cluster on the right in Figure shows the distribution of the strength of anchoring between the second item and

27 End anchoring in STM the start marker (i.e., the value of the start marker at the second within-group position). These values are all fairly high (the upper bound is ), indicating some strong anchoring between the start marker and its immediately adjacent internal item. In contrast, the anchoring values giving the strength of anchoring between the end of the group and the third item (i.e., the internal item closest to the end of the group) are clustered near 0 on the left. Although a few of these values are above 0, the largest end anchoring value is still far the below the smallest start anchoring strength seen across the participants. Finally, the primacy+s-m-e model (ˆα =.8; ˆτ =.; ˆγ =.; R init =.; mix =.8) performs poorly with respect to the other three models; this model never provides a superior fit and has an average BIC weight indistinguishable from 0 (Table ). Figure shows that the SPCs predicted by the model fail to quantitatively capture the empirical SPCs. At a finer level of detail, Figure shows that the model s predicted transposition matrices deviate markedly from the data, and that specifically the model overpredicts the confusion of items internal to groups (particularly items and ), as well as underpredicting performance on the fourth position. In summary, the modeling has replicated the fits to Henson s (999b) data: the restricted end model gives a satisfactory account of serial recall of grouped lists, and the addition of an extra parameter in the extensive end marker model is not warranted by the data. One important difference from Henson s (999b) experiment was that the use of a grouping structure allowed a detailed analysis of transpositions involving items internal to groups, which should have revealed any evidence for an extensive end marker having its influence across the entire group. One possibility is that extensive end anchoring may be a general property of memory for serial order, but that this experiment was not powerful enough to test differences between the restricted end and extensive end models (despite the large differences across other model comparisons). Accordingly, an additional experiment was